Incremental parameterized-kernel fractional operators: theory, novel Banach spaces, and machine learning applications

This paper presents a new class of Incremental Parameterized–Kernel (IPK) fractional derivative and integral operators that improve modeling of complex dynamical systems. Two specially constructed increment-based left- and right-sided integrals are iterated to prove Cauchy-type formulas which lead to the generalized IPK fractional operators. Essential properties such as linearity, boundedness, semigroup structure, and continuity are verified, and it is shown that the associated operator spaces constitute Banach spaces. Several special cases of the parameters provide classical fractional forms such as Riemann–Liouville, Hadamard, and Katugampola operators. A comparative study is also conducted that highlights the enhanced memory behavior and adaptability of the IPK operators relative to traditional models. Additionally, the newly-defined operators are implemented in a neural network training framework, where IPK-based fractional gradients improve learning performance and demonstrate computational feasibility. These results underscore the ability of the IPK operators for applications in mathematical modeling, and AI systems with memory-dependent dynamics.